Well HTTP authentication might be nice if it didn't have a serious drawback: it's just not possible to end a session. Mozilla used to be the only browser that I know of that ever had a logout button for HTTP authentication. It was removed in Firefox to simplify the user interface. You can still add it for example as part of the web developer extensions, but no normal user would have that.
Also it's not possible to end a session from the server side, since the browser is sending valid credentials with every request. It's just a NO GO from a security perspective.